Abstract

In this work, a nonlinear deterministic model for schistosomiasis transmission including delays with two general incidence functions is considered. Rigourous mathematical analysis is done. We show that the stability of the disease-free equilibrium and the existence of an endemic equilibrium for the model are stated in terms of key thresholds parameters known as basic reproduction number R0. This study of the dynamic of the model is globally asymptotically stable if R0≤1, and the unique endemic equilibrium is globally asymptotically stable when R0>1. Some numerical simulations are provided to support the theoretical result with respect to R0 in this paper.

Keywords

1 Introduction

Schistosomiasis is a serious health problem in developing countries. Indeed, despite the remarkable achievements in schistosomiasis control over the past five decades, there are about 240 million people infected worldwide, and more than 700 million people live in endemic areas [1]. There are two patterns of schistosomiasis. We note the urinary schistosomiasis and the intestinal schistosomiasis. The first one is caused by Schistosoma haematobium, when the second is caused by any of the organisms Schistosoma intercalatum, Schistosoma mansoni, Schistosoma japonicum and Schistosoma mekongi. Mathematical modeling of schistosomiasis transmission can help in the development of the strategies for control. Thus, several mathematical models for this disease have been done (see [2–13] and the references therein). In [11], a discrete delay model for the transmission is studied. The delay appears in the incidence term including masse action SI (S: susceptible, I: infectious). It appears that the incidence function form is determinative in the study of the model system. Then, changing the form of the incidence can potentially change the behaviour of the system. In this paper, a mathematical model is derived with a bounded delay distributed and two general incidence functions term f and g. The model described here considers two population hosts, humans and snails, and is structured as follows: Susceptible (uninfected) and infectious humans and susceptible (uninfected) and infected snails. The paper is organized as follows. In Section 2, we present the mathematical model, and we study the mathematical properties of the model system. In Section 3, we derive some results about the basic reproduction number, the disease-free equilibrium and the endemic equilibrium. Section 4 is devoted to the global stability of the disease-free equilibrium. In Section 5, we study the global stability of the endemic equilibrium. Section 6 is devoted to numerical simulation. Finally, in Section 7, we end by a conclusion.

2 The mathematical model

In this section, we derive a mathematical model for the spread of schistosomiasis. Here, we consider human and snail populations. We assume that all newborns are susceptible, and that the infection does not result in death of human and snail populations. Further, it is assumed that a susceptible host became infected only by contact with water, in which there exist cercariae from infected snails, and a susceptible snail became infected by contact with miracidia coming from parasite eggs released in feces and urine of infected hosts (see Figure 1 below).

Figure 1

Transfer diagram for the mathematical model.

We denote by

Hs the susceptible (uninfected) human population size;

Hi the infected human population size (infectious humans);

Ss the susceptible (uninfected) snail population size;

Si the infected and shedding snail population size (shedding snail population size).

We also denote by

Λh the recruitment rate of susceptible humans;

Λs the recruitment rate of susceptible snails;

dh the per capita natural death rate of humans;

ds the per capita natural death rate of snails;

τ1 the transit time from cercaria in water to schistosomule in a human host;

τ2 the transit time from parasite eggs to miracidia to infect a snail;

k1 and k2 the Lebesgue integrable functions, which give the relative infectivity of snails and humans (respectively) of different infection ages.

Note that the support of k1 and k2 has a positive measure in any open interval having supremum h, so that the interval of integration is not artificially extended by concluding with an interval, for which the integral is automatically zero. On the other hand, we choose in the model two real numbers α and γ, so that ∫0hk1(τ1)dτ1=1 and ∫0hk2(τ2)dτ2=1.

As general as possible, the incidence functions f and g must satisfy technical conditions. Thus, we assume that

H1 f and g are non-negative C1 functions on the non-negative quadrant,

H2 for all (HS,Hi,Ss,Si)∈R+4, f(Hs,0)=f(0,Si)=0 and g(Ss,0)=g(0,Hi)=0.

Remark 2.1f and g are two incidence functions, which explain the contact between two species. Therefore, f and g are non-negative. Note also that when there is no one infected in the human and snail populations, then the incidence functions are equal to zero. The incidence functions are also equal to zero, when there is no one susceptible in the human and snail populations.

Let us denote by f1 and f2 the partial derivatives of f with respect to the first and to the second variable and g1 and g2 those of g with respect to the first and to the second variable. For mathematical simplicity, we shall make now a simplification that will allow us to carry out an analysis, namely we assume that the disease-induced death rate is neglected. Using the notations Hτ2i=Hi(t−τ2) and Sτ1i=Si(t−τ1), the model equations are given as follows

where ν1,ν2∈C([−h,0],R+). We also define the sup norm on C([−h,0],R+) as ∥νi∥=supθ∈[−h,0]νi(θ), i=1,2. Standard theory of functional differential equations (see [14]) can be used to show that solutions of system (2.1) exist and are differentiable for all t>0.

The delay is inspired by the life history of the schistosomiasis. Indeed, it is possible that some hosts or intermediate hosts (snails) die due to natural death during the incubation period, respectively (see [11]).

Then 〈X(x)|∇L(x)〉=−Λh<0. This proves that {Hs≥0} is positively invariant. Similarly, we prove that {Hi≥0}, {Ss≥0}, {Si≥0} are positively invariant. Then {(Hs,Hi,Ss,Si)∈R4:Hs≥0,Hi≥0,Ss≥0,Si≥0} is positively invariant for system (2.1). □

Therefore, the model is mathematically well posed and epidemiologically reasonable since all the variables remain non-negatives for all t>0.

Theorem 2.4Assume thatH(t)=Hs(t)+Hi(t)andS(t)=Ss(t)+Si(t).

There existsϵ≥0such as all feasible solutions of model system (2.1) enter the set

Combining the two first equations and the two last equations of (3.17) gives

Hs∗=Λh−dhHi∗dhandSs∗=Λs−dsSi∗ds.

(3.18)

Let

ϕ1(Hi∗,Si∗)=αf(Λh−dhHi∗dh,Si∗)−dhHi∗

(3.19)

and

ϕ2(Hi∗,Si∗)=γg(Λs−dsSi∗ds,Hi∗)−dsSi∗.

(3.20)

Now, we define the continuous function ϕ by

ϕ(Hi∗,Si∗)=(ϕ1(Hi∗,Si∗),ϕ2(Hi∗,Si∗)).

(3.21)

Hence, it follows that any solution of equation ϕ=0 in the set (0,Λhdh)×(0,Λsds) corresponds to an equilibrium, with Hs∗,Hi∗,Ss∗,Si∗>0, that is an equilibrium. Since H2 holds, then ϕ(0,0)=0 and ϕ(Λhdh,Λsds)≤0. Then the sufficient condition for equation ϕ=0 to have a solution in (0,Λhdh)×(0,Λsds) is that ϕ increasing at 0. This implies that an endemic equilibrium exists if

with equality only if Si=0 and Hi=0. According to LaSalle’s extension to Lyapunov’s method [19], the limit set of each solution is contained in the largest invariant set, for which Si=0 and Hi=0, which is the singleton {E0}. This means that the disease-free equilibrium E0 is globally asymptotically stable on Γϵ. □

5 Global stability of the endemic equilibrium

In this section, we assume that f and g satisfies the conditions

H5 for all (Hs,Hi,Ss,Si)∈R+4, 1≤f(Hs,Sτ1i)f(Hs,Si∗)≤SiSi∗ and 1≤g(Ss,Hτ2i)g(Ss,Hi∗)≤HiHi∗,

H6 for all Hs,Ss>0, sign(f(Hs,Si∗)−f(Hs∗,Si∗))=sign(Hs−Hs∗) and sign(g(Ss,Hi∗)−g(Ss∗,Hi∗))=sign(Ss−Ss∗).

for all (Hs,Hi,Ss,Si)∈Γϵ with equality only for Hs=Hs∗, Hi=Hi∗, Ss=Ss∗ and Si=Si∗. Hence, the endemic equilibrium E∗ is the only positively invariant set of system (2.1) contained in {(Hs,Hi,Ss,Si)∈R+4;Hs=Hs∗,Hi=Hi∗,Ss=Ss∗,Si=Si∗}. Then it follows that E∗ is globally asymptotically stable on Γϵ (see [19]). □

6 Numerical simulation

In this section, we derive the computation work that supports our study. In this computation, the functions f and g are chosen as follows: f(Hs,Si)=HsSi and g(Ss,Hi)=SsHi (mass action). Two different cases of computational simulations are studied: in the first case (see Figure 2), R0≤1, while in the second case (see Figure 3), R0>1. The parameters values used are the following (see [20]): Λh=8, Λs=25, dh=0.014, ds=0.2. We also use the delays parameters τ1=τ2=5. For the first case, we use α=0.00000027 and γ=0.000004, which give R0<1. In the second case, we use α=0.00027 and γ=0.004 to get R0>1.

Figure 2

Case, whereR0<1.

Figure 3

Case, whereR0>1.

7 Conclusion

In this paper, a deterministic model of transmission of schistosomiasis with two general nonlinear incidence functions including distributed delay is derived. The global behaviour of the model system was studied. We proved that, if R0≤1 holds, then the disease-free equilibrium is globally asymptotically stable, which implies that the disease fades out from the population. If R0>1, then there exists a unique endemic equilibrium which is globally asymptotically stable, and this implies that the disease will persist in the population. This result suggests that the latent period in infection affects the prevalence of schistosomiasis, and it is an effective strategy on schistosomiasis control to lengthen in prepatent period on infected definitive hosts by drug treatment, for example.

Threshold analysis of the basic reproduction number shows that the use of public health education campaign could have positive, more determinant impact on the control of the schistosomiasis. Overall, an effective education campaign which focuses on drug treatment with reasonable coverage level could be helpful for countries concerned with the disease.

Declarations

Acknowledgements

The authors express their deepest thanks to the editor and an anonymous referee for their comments and suggestions on the article.

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